Dynamical Manin–Mumford conjecture for polarized endomorphisms

Let $f:X\to X$ be a polarized endomorphism of a smooth projective variety over a field of characteristic zero, and let $Z\subset X$ be a subvariety containing a Zariski dense set of preperiodic points. Dynamical Manin–Mumford conjecture. Then either $Z$ is preperiodic or $Z$ is special: it is contained in a subvariety $Y$ that is both $f^n$- and $\psi$-invariant for some $n\geq 1$, where $\psi$ is another polarized endomorphism commuting with $f^n$ on $Y$, and $Z$ is preperiodic under $\psi$. This is a dynamical analogue of the Manin–Mumford conjecture; the relative and higher-dimensional forms remain open in general.